Limitations on how far one can travel in the universe

Someone once incorrectly told me that, given the speed of light is the speed limit of the universe, aliens would have to live for hundreds of years if they are to travel distances of hundreds of light years to reach Earth.

In a "special relativistic" and non-expanding universe however, this is not the case. As velocity approaches the speed of light, say $v = 0.999c$, then we have

$$\gamma = \frac{1}{\sqrt{1-\frac{(0.999c)^2}{c^2}}} = \frac{1}{\sqrt{1-\frac{0.998001c^2}{c^2}}} = 22.37$$

Let us assume that an alien wishes to travel 100 light years from his planet to Earth. If the alien is travelling at $v = 0.999c$, he will observe the distance between his planet and the Earth to contract, and will measure the contracted distance to be:

$$\text{Distance} = \frac{100 \; \mathrm{ly}}{\gamma} = \frac{100 \; \mathrm{ly}}{22.37} = 4.47 \; \text{light years}$$

The Alien will be able to travel this distance in a time of :

$$\text{Time} = \text{distance}/\text{speed} = 4.47/0.999 = 4.47 \; \text{years}$$

It is easy to show that as the alien's speed increases, the time taken to travel the 100 light year distance approaches 0. It can thus be shown that thanks to length contraction and time dilation of special relativity, all parts of a special relativistic universe are accessible to an observer with a finite life time.

We however don't live in a purely special relativistic universe. We live in an expanding universe. Given the universe is expanding, are some parts of the universe no longer theoretically accessible to observers with finite life times?

Your question can be translated into "if right now we would send a powerful omnidirectional light pulse from earth into space, would there be galaxies that never see this light pulse?"

The answer is "yes". Due to the accelerated expansion of the universe, as described by the lambda-CDM model, only galaxies currently less than about 16 billion light years (the difference between the cosmological event horizon and the current distance to the particle horizon) away from us will at some time observe the light pulse.

A nice visual representation of this can be found in figure 1 of this publication.

• Hypothetically supposing we could choose to send the light pulse any time after the Big Bang happened, would the answer ever be "no" to your translated question? Assume you can avoid the questionable periods in the leading moments of the Big Bang, and perhaps limit to any time after the first hydrogen atom formed. Mainly I want to know if it would be possible that some light pulse in the past could potentially reach every galaxy, or if it's just a larger but still finite limit? – codefactor Jan 4 '14 at 7:41
• Actually, in your link (which I may be reading wrong) indicates the limit at t=0 (beginning of time) would be 46 billion light years. Am I reading that right? If so then the answer to my question is that your translated question would still be "yes" regardless of when the light pulse occurred. – codefactor Jan 4 '14 at 7:51

The short answer is yes, there are often parts of the universe an observer can never reach due to expansion. Here are some details:

You're absolutely right that curved spacetime is involved in this question. First let me state an obvious point. Even in flat spacetime, there are always events a given observer cannot reach. Namely, if a bomb sits 1 meter away from you and it is set to explode in $10^{-15}$ seconds (in your initial reference frame), you can't reach it because the speed of light is too slow. The event is said to be outside of your causal future.

However, you can still reach the location where the bomb was sitting before it exploded. It can be done in arbitrarily little time on your clock by moving arbitrarily close to $c$. That is (roughly) the sense in which special relativity allows you to "reach any place in the universe" in arbitrarily small proper time.

In curved spacetime (which describes the expanding universe) a new problem arises. Consider the bomb example discussed above. We'd like to know if you can reach "the location where the bomb sat" in a small amount of proper time. However, this "location" is not a well-defined concept in GR. In completely general spacetimes, the best you can say is that you can't reach events that are outside of your causal future (like the unexploded bomb).

The good news is that in the special case of cosmology there is a notion of the location where the bomb sat. The idea is to imagine that the universe is full of galaxies and to say that two events occur at the same (comoving) location if they are close to the same galaxy. The fact is that in many cosmological solutions, there will be galaxies that follow a path through spacetime that is outside of your causal future. You can never reach them (you may be able to see them though).

An example: our universe is becoming closer and closer to an exponentially expanding universe. In that case, there is a certain distance (16 Glyr for us) such that any observer farther than that away can never reach you. You can think of this as being because the expansion is too fast for the observer to get back to earth.

• +1 for "this location is not a well-defined concept in GR" and then the cosmological special case! – twistor59 Dec 22 '12 at 14:29